Method for employing SEC-FTIR data to predict mechanical properties of polyethylene

ABSTRACT

The present invention provides several methods of determining values of physical or chemical properties of polymers. In these methods, at least two polymer training samples are provided. Characteristics of the polymer microstructures of the training samples are correlated with values of physical or chemical properties of the training samples. These correlations are subsequently applied to the respective characteristics of polymer test samples in order to determine the values of physical or chemical properties of the test samples.

BACKGROUND OF THE INVENTION

The present invention is directed to methods of determining values ofphysical or chemical properties of polymers. Traditionally, in order todetermine a value of a specific physical or chemical property, a certainquantity of the particular polymer resin was needed to fabricate anarticle or a test specimen, and then the resulting article or testspecimen was subsequently tested via the prescribed analytical testprocedure to determine the value of the physical or chemical property.This procedure is cumbersome not only due to the time required forfabricating the article or test specimen, but also the time required toperform the respective analytical test procedure. Further, thetraditional method, depending upon the particular test, could requirelarge amounts of polymer, often more than could be produced insmall-scale research laboratory or pilot plant apparatus.

Hence, there exists a need for a method of determining a value of adesired chemical or physical property of a test sample that requiresonly a small amount of polymer for analysis. Further, this method shouldprovide for the value of the physical or chemical property of the testsample to be determined without fabricating an article or a testspecimen. Still further, this method should allow determination of thevalue of the physical or chemical property of the test sample withoutperforming the analytical test for the physical or chemical property.

In another aspect, there is a need for a method of rapidly determining avalue of a desired chemical or physical property of a test sample. It isbeneficial to determine the value of the chemical or physical propertywithout having to invest the time and expense needed to fabricatearticles or test specimens for subsequent analysis, nor to invest thetime and expense needed to perform the analytical test procedure, whichdependent upon the specific test, could take days or weeks to determinethe value of the respective property. The cycle time of resin design andproduct development to achieve a desired physical or chemical propertyattribute could be greatly reduced with a method that provided rapidfeedback using only a small quantity of a polymer test sample.Accordingly, the methods of the present invention are directed above.

BRIEF SUMMARY OF THE INVENTION

The present invention discloses a method of determining values ofphysical or chemical properties of polymers. In one aspect, the presentinvention provides a method of determining a value of a physical orchemical property of at least one polymer test sample. The at least onepolymer test sample has a molecular weight distribution (MWD) profileand a short chain branching distribution (SCBD). In this aspect, themethod of determining a value of a physical or chemical property of theat least one polymer test sample comprises:

a) providing at least two polymer training samples, each training samplehaving a MWD profile, a SCBD, and a known value of the respectivephysical or chemical property;

b) determining at least two weighted cross terms at respective molecularweights along the MWD profile and the SCBD, for each of the at least twopolymer training samples and the at least one polymer test sample, eachweighted cross term being determined as the multiplication product of:

-   -   (1) the respective molecular weight;    -   (2) the weight fraction at the respective molecular weight; and    -   (3) the number of short chain branches per 1000 carbon atoms at        the respective molecular weight;

c) plotting each weighted cross term versus the logarithm of therespective molecular weight for each of the at least two polymertraining samples and the at least one polymer test sample;

d) determining the respective area under each curve in step c);

e) correlating the respective areas for each of the at least two polymertraining samples in step d) with the known value of the respectivephysical or chemical property; and

f) applying the correlation of step e) to the respective area of the atleast one polymer test sample to determine the value of the physical orchemical property of the at least one polymer test sample.

In another aspect of the present invention, the method is directed to asingle-point determination. That is, the weighted cross term isdetermined at a single molecular weight. In this aspect, the presentinvention provides a method of determining a value of a physical orchemical property of at least one polymer test sample. The at least onepolymer test sample has a molecular weight, a weight fraction at therespective molecular weight, and a number of short chain branches per1000 carton atoms at the respective molecular weight. This methodcomprises:

a) providing at least two polymer training samples, each training samplehaving a molecular weight, a weight fraction at the respective molecularweight, a number of short chain branches per 1000 carton atoms at therespective molecular weight, and a known value of the respectivephysical or chemical property;

b) determining a weighted cross term at the respective molecular weightfor each of the at least two polymer training samples and the at leastone polymer test sample, each weighted cross term being determined asthe multiplication product of:

-   -   (1) the respective molecular weight;    -   (2) the weight fraction at the respective molecular weight; and    -   (3) the number of short chain branches per 1000 carbon atoms at        the respective molecular weight;

c) correlating the respective weighted cross terms for each of the atleast two polymer training samples in step b) with the known value ofthe respective physical or chemical property; and

d) applying the correlation of step c) to the weighted cross term of theat least one polymer test sample to determine the value of the physicalor chemical property of the at least one polymer test sample.

In yet another aspect of the present invention, a chemometric method canbe used to determine a value of a physical or chemical property of atleast one polymer test sample. The at least one polymer test sample hasa MWD profile and a SCBD. This method comprises:

a) providing at least two polymer training samples, each training samplehaving a MWD profile, a SCBD, and a known value of the respectivephysical or chemical property;

b) determining at least two weighted cross terms at respective molecularweights along the MWD profile and the SCBD, for each of the at least twopolymer training samples and the at least one polymer test sample, eachweighted cross term being determined as the multiplication product of:

-   -   (1) the respective molecular weight;    -   (2) the weight fraction at the respective molecular weight; and    -   (3) the number of short chain branches per 1000 carbon atoms at        the respective molecular weight;

c) by chemometric analysis, defining a mathematical relationship betweenthe values of the respective physical or chemical property and theweighted cross terms for each of the at least two polymer trainingsamples; and

d) applying the mathematical relationship of step c) to the respectiveweighted cross terms of step b) for the at least one polymer test sampleto determine the value of the physical or chemical property of the atthe least one polymer test sample.

In a different aspect, the present invention provides a method ofdetermining a value of a physical or chemical property of at least onepolymer test sample using tie molecule probabilities. For example, theat least one polymer test sample has a composite density, a MWD profile,and a SCBD. The method of determining a value of a physical or chemicalproperty of the at least one polymer test sample comprises:

a) providing at least two polymer training samples, each training samplehaving a composite density, a MWD profile, a SCBD, and a known value ofthe respective physical or chemical property;

b) determining at least two density terms at respective molecularweights along the MWD profile and the SCBD, for each of the at least twopolymer training samples and the at least one polymer test sample, eachdensity term being determined using the composite density, the MWDprofile, and the SCBD;

c) determining a respective melting temperature from each density termin step b);

d) determining a respective probability for tie molecule formation fromeach melting temperature in step c);

e) determining a respective weighted tie molecule probability, eachweighted tie molecule probability being determined as the multiplicationproduct of:

-   -   (1) the weight fraction at the respective molecular weight; and    -   (2) the probability for tie molecule formation in step d) at the        respective molecular weight;

f) plotting each weighted tie molecule probability versus the logarithmof the respective molecular weight for each of the at least two polymertraining samples and the at least one polymer test sample;

g) determining the respective area under each curve in step f);

h) correlating the respective areas for each of the at least two polymertraining samples in step g) with the known value of the respectivephysical or chemical property; and

i) applying the correlation of step h) to the respective area of the atleast one polymer test sample to determine the value of the physical orchemical property of the at least one polymer test sample.

A further aspect of the present invention is a single-point method whichutilizes tie molecule probabilities. That is, the weighted tie moleculeprobability is determined at a single molecular weight. In this aspect,the present invention provides a method of determining a value of aphysical or chemical property of at least one polymer test sample. Theat least one polymer test sample has a composite density, a molecularweight, and a weight fraction at the respective molecular weight. Thismethod comprises:

a) providing at least two polymer training samples, each training samplehaving a composite density, a molecular weight, a weight fraction at therespective molecular weight, and a known value of the respectivephysical or chemical property;

b) determining a minimum molecule length for a tie molecule(2L_(c)+L_(a)) using the composite density for each of the at least twopolymer training samples and the at least one polymer test sample;

c) determining a respective probability for tie molecule formation atthe respective molecular weight from each 2L_(c)+L_(a) of step b);

d) determining a respective weighted tie molecule probability, eachweighted tie molecule probability being determined as the multiplicationproduct of:

-   -   (1) the weight fraction at the respective molecular weight; and    -   (2) the probability for tie molecule formation in step c) at the        respective molecular weight;

e) correlating the respective weighted tie molecule probability for eachof the at least two polymer training samples in step d) with the knownvalue of the respective physical or chemical property; and

f) applying the correlation of step e) to the weighted tie moleculeprobability of the at least one polymer test sample to determine thevalue of the physical or chemical property of the at least one polymertest sample.

DEFINITIONS

The following definitions are provided in order to aid those skilled inthe art in understanding the detailed description of the presentinvention.

dW/d Log M Weight fraction. ESCR Environmental Stress Crack Resistance,ASTM D1693. FNCT Full Notched Creep Test, ISO 16770. FTIR FourierTransform-Infrared spectrophotometry. M or MW Molecular weight. MWDMolecular weight distribution. M_(n) Number average MW. M_(w) Weightaverage MW. NDR Natural Draw Ratio, ASTM D638. NPT Notched Pipe Test,ISO 13479. PENT Pennsylvania Notched Test, ASTM F1473, measured inhours. PSP1 Primary structure parameter 1; calculated by dividing thearea under the curve of the plot of the weighted cross term versusmolecular weight by 100,000. PSP2 Primary structure parameter 2;calculated by multiplying the area under the curve of the plot of theweighted tie molecule probability versus molecular weight by 100. P_(TM)or P Probability for tie molecule formation. SCBD Short chain branchingdistribution. SCB's Number of short chain branches per 1000 carbonatoms. SEC Size Exclusion Chromatography; also referred to as GelPermeation Chromatography (GPC). SP-NCTL Single Point Notched ConstantTensile Load, ASTM D5397, 30% yield. Weighted Multiplication product ofa molecular weight, the weight cross fraction at the respectivemolecular weight, and the number of short chain term branches per 1000carbon atoms at the respective molecular weight; also shown as M *dW/d(Log M) * SCB. Weighted Multiplication product of the weightfraction at tie molecule a molecular weight and the probability for tiemolecule formation at the probability respective molecular weight; alsoshown as dW/d(Log M) * P_(TM). 2L_(c) + L_(a) Minimum molecule lengthfor a tie molecule; L_(c) is the crystalline lamella thickness and L_(a)is the amorphous layer thickness.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 presents a plot of the molecular weight distribution (MWD)profile and the short chain branching distribution (SCBD) for anexemplary polyolefin copolymer.

FIG. 2 presents a plot of the weighted cross term versus the logarithmof the molecular weight for an exemplary polyolefin copolymer.

FIG. 3 presents a plot of Log PENT (in hours) at 2.4 MPa versus primarystructure parameter 1 (PSP1) for two exemplary series of polyolefinpolymers.

FIG. 4 presents a plot illustrating an empirical correlation ofcomposite density versus the logarithm of weight-averaged molecularweight for an exemplary homopolymer.

FIG. 5 presents a plot of the MWD profile and the respective homopolymerand copolymer density terms across the MWD profile for an exemplarypolyolefin copolymer.

FIG. 6 presents a plot illustrating an empirical correlation of meltingtemperature versus density for polyethylene polymers.

FIG. 7 presents a plot of 2L_(c)+L_(a) (measured in nm) as a function ofdensity for one aspect of a method of the present invention comparedwith reported literature values.

FIG. 8 presents a plot of the weighted tie molecule probability versusthe logarithm of the molecular weight for two exemplary polyolefincopolymers.

FIG. 9 presents a plot of the molecular weight distribution (MWD)profile and the short chain branching distribution (SCBD) for bimodalpolyethylene polymer BM-1.

FIG. 10 presents a plot of the molecular weight distribution (MWD)profile and the short chain branching distribution (SCBD) for bimodalpolyethylene polymer BM-2.

FIG. 11 presents a plot of the molecular weight distribution (MWD)profile and the short chain branching distribution (SCBD) for bimodalpolyethylene polymer BM-3.

FIG. 12 presents a plot of the molecular weight distribution (MWD)profile and the short chain branching distribution (SCBD) for bimodalpolyethylene polymer BM-4.

FIG. 13 presents a plot of the molecular weight distribution (MWD)profile and the short chain branching distribution (SCBD) for bimodalpolyethylene polymer BM-5.

FIG. 14 presents a plot of the molecular weight distribution (MWD)profile and the short chain branching distribution (SCBD) for bimodalpolyethylene polymer BM-6.

FIG. 15 presents a plot of the molecular weight distribution (MWD)profile and the short chain branching distribution (SCBD) for bimodalpolyethylene polymer BM-7.

FIG. 16 presents a plot of the weighted cross term versus the logarithmof the molecular weight for bimodal polyethylene polymer trainingsamples BM-1 to BM-7.

FIG. 17 presents a plot of Log PENT (in hours) at 2.4 MPa versus PSP1for bimodal polyethylene polymer training samples BM-2 to BM-7.

FIG. 18 presents a plot of the weighted cross term versus the logarithmof the molecular weight for polyethylene polymer training samplesproduced using a chromium-based catalyst.

FIG. 19 presents a plot of Log PENT (in hours) at 2.4 MPa versus PSP1for polyethylene copolymer training samples produced using achromium-based catalyst.

FIG. 20 presents a plot of Log PENT (in hours) at 2.4 MPa versusweighted cross terms divided by 100,000 for bimodal polyethylene polymertraining samples.

FIG. 21 presents a plot of predicted PENT (in hours) at 2.4 MPa usingchemometric analysis versus measured PENT for various polyethylenepolymer training samples.

FIG. 22 presents a plot of predicted PENT (in hours) at 2.4 MPa usingchemometric analysis versus measured PENT for various polyethylenepolymer training samples.

FIG. 23 presents a plot of Log PENT (in hours) at 2.4 MPa versus PSP2for polyethylene polymer training samples produced using differentcatalyst systems.

FIG. 24 presents a plot of Log SP-NCTL (in hours) versus PSP2 forpolyethylene polymer training samples produced using different catalystsystems.

FIG. 25 presents a plot of NDR versus PSP2 for polyethylene polymertraining samples produced using different catalyst systems.

FIG. 26 presents a plot of Log PENT (in hours) at 2.4 MPa versusweighted tie molecule probability for bimodal polyethylene polymertraining samples.

DETAILED DESCRIPTION OF THE INVENTION

The present invention is directed to a method of determining values ofphysical or chemical properties of polymers. Exemplary physical orchemical properties of interest in the present invention include, butare not limited to, PENT, ESCR, SP-NCTL, NPT, FNCT, NDR, izod impact,dart impact, Charpy impact, puncture resistance, or Elmendorf tearstrength. The PENT value can be determined at different conditions, suchas, for example, 2.4 MPa, 3.0 MPa, or 3.8 MPa.

The method of the present invention is applicable to all classes ofpolymers, although the method is particularly well suited forsemi-crystalline polymers. The invention will be described in connectionwith polyolefins, particularly polyethylene homopolymers and copolymers.It is understood that the present invention is not limited to theaspects and examples outlined herein, and that the present inventionincludes all alternatives, modifications, and equivalents as may beincluded within the spirit and the scope of the specification and claimsthat follow.

Differences in the polymer microstructure, such as the MWD profile andthe SCBD of a given polymer, can influence the resulting physical orchemical properties of that polymer. Thus, in one aspect of the presentinvention, the method described herein can determine the values ofphysical or chemical properties of test samples using knowledge of therespective test samples MWD profile and SCBD. In another aspect, amethod in accordance with this invention further requires a compositeresin density of the polymer test sample.

The MWD profile of a polymer can be provided by any means known to oneor ordinary skill in the art. A non-limiting example of an analyticaltechnique to determine the MWD profile of a polymer is SEC or GPC.Inherently, as used in this disclosure, the MWD profile of a polymer canprovide, among other data, the MWD data and associated weight fractionat each MW, including common terms useful in the art such as M_(w) andM_(n).

Similarly, the SCBD of a polymer can be provided by any means known toone of skill in the art. Techniques could include, but are not limitedto, temperature rising elution fractionation (TREF), nuclear magneticresonance (NMR), and SEC-FTIR. Inherently, as used in this disclosure,the SCBD of a polymer can provide the number of short chain branches per1000 carbon atoms (SCB's) at each MW across the MWD profile.

One method to provide both the MWD profile and the SCBD of a polymer isSEC-FTIR using chemometric analysis, described in U.S. Pat. No.6,632,680 and U.S. patent application Ser. No. 10/463,849, thedisclosures of which are incorporated in their entirety by thisreference. An advantage of SEC-FTIR as it relates to the methods of thepresent invention is the small quantity of the polymer training samplesand of the polymer test samples that are required for analysis todetermine the MWD profile and the SCBD. In one aspect of the presentinvention, less than about 5 grams each of the respective test ortraining samples is provided for determination of the MWD profile andthe SCBD. In another aspect, less than about 1 gram is provided foranalysis. In a further aspect, less than about 100 milligrams isprovided. Further, another advantage of the present invention is thequick determination of a physical or chemical property of a test sampleas compared to conventional testing and measurement methods. Using atechnique such as SEC-FTIR in combination with the methods of thepresent invention, the value of a physical or chemical property can bedetermined in less than three hours. In another aspect, the value of aphysical or chemical property can be determined in less than two hours.In yet another aspect, the value of a physical or chemical property canbe determined in less than one hour.

Composite resin density likewise can be determined by any means known toone of ordinary skill in the art. Analytical techniques include, but arenot limited to, refractive index or molded density per ASTM D 1238. Thecomposite resin density is the density of the polymer as a whole, acrossall molecular weights.

While methods are described in terms of “comprising” various steps, themethods can also “consist essentially of” or “consist of” the varioussteps.

Primary Structure Parameter 1 (PSP1)

One aspect of the present invention provides a method of determining avalue of a physical or chemical property of at least one polymer testsample, the at least one polymer test sample having a molecular weightdistribution (MWD) profile and a short chain branching distribution(SCBD). The MWD profile and the SCBD can be determined via anyanalytical technique known to one of ordinary skill in the art, such asSEC-FTIR. The polymer test sample can also be referred to as anexperimental sample or an unknown sample. The method of determining avalue of a physical or chemical property of the at least one polymertest sample comprises:

a) providing at least two polymer training samples, each training samplehaving a MWD profile, a SCBD, and a known value of the respectivephysical or chemical property;

b) determining at least two weighted cross terms at respective molecularweights along the MWD profile and the SCBD, for each of the at least twopolymer training samples and the at least one polymer test sample, eachweighted cross term being determined as the multiplication product of:

-   -   (1) the respective molecular weight;    -   (2) the weight fraction at the respective molecular weight; and    -   (3) the number of short chain branches per 1000 carbon atoms at        the respective molecular weight;

c) plotting each weighted cross term versus the logarithm of therespective molecular weight for each of the at least two polymertraining samples and the at least one polymer test sample;

d) determining the respective area under each curve in step c);

e) correlating the respective areas for each of the at least two polymertraining samples in step d) with the known value of the respectivephysical or chemical property; and

f) applying the correlation of step e) to the respective area of the atleast one polymer test sample to determine the value of the physical orchemical property of the at least one polymer test sample.

At least two different polymer training samples are provided inaccordance with the methods of the present invention. Training samplescan also be referred to as control samples. Each polymer training samplehas a known MWD profile and a known SCBD. The MWD profile and the SCBDcan be determined using a technique such as SEC-FTIR, as mentionedabove. For an exemplary polyolefin copolymer, FIG. 1 illustrates the MWDprofile and the SCBD of the copolymer. The x-axis is the logarithm ofthe molecular weight, the left hand side y-axis is the weight fractionat each molecular weight, and the right hand side y-axis is the numberof short chain branches per 1000 carbon atoms (SCB's). Graphical data asillustrated in FIG. 1 can be provided for both the at least two polymertraining samples and the at least one polymer test sample.

Further, each polymer training sample has a known value of therespective physical or chemical property that is of interest. Forinstance, at least two polymer training samples can be provided thateach have a known value of PENT at 2.4 MPa, the PENT value having beendetermined previously using the respective analytical test for PENT.

At least two polymer training samples are used in the methods of thepresent invention. Alternatively, however, at least three trainingsamples, at least four training samples, at least five training samples,at least ten training samples, at least fifteen training samples, or atleast twenty training samples, can be used. There is no specific upperlimit on the number of different training samples that can be used withthe present invention. It is beneficial that the training samplesresemble the polymer test sample, and that the range covered by thetraining samples encompass the polymer test sample, but this is notrequired. Further, the training samples can include duplicate orredundant samples to include the impact of experimental error in thetests for MWD profile and SCBD, and of the respective physical orchemical property of interest.

For each of the at least two polymer training samples and the at leastone polymer test sample, weighted cross terms can be determined usingthe respective data for each sample as exemplified in FIG. 1. A weightedcross term is the multiplication product of a molecular weight, theweight fraction at the respective molecular weight, and the number ofshort chain branches per 1000 carbon atoms at the respective molecularweight (i.e., M*dW/d(Log M)*SCB's). While not intending to be bound bythis theory, Applicants submit that one factor related to polymers withimproved toughness (e.g., tear, impact, puncture, or stress crackresistance) is the presence of more short chain branching at highermolecular weights. That is, the presence of high levels of branching atlower molecular weights does not contribute significantly to improvedtoughness (e.g., in properties such as PENT, SP-NCTL, dart impact, andthe like). Rather, high short chain branch content present on the highmolecular weight end of the MWD profile can contribute significantly toimproved toughness of polymers. Again, not intending to be bound by thistheory, Applicants believe that the weighted cross term, as definedabove, captures the impact of having a higher branching content on thehigher molecular weight fraction of the polymer.

In accordance with one aspect of the present invention, at least twoweighted cross terms at respective molecular weights along the MWDprofile and the SCBD are required. Each weighted cross term is plottedversus the logarithm of the respective molecular weight, for each of theat least two polymer training samples and the at least one polymer testsample. This is illustrated in FIG. 2 for an exemplary polyolefincopolymer. Although at least two weighted cross terms are required, itis beneficial to have many more cross terms across the range ofmolecular weights to form a curve as illustrated in FIG. 2. Hence, inaccordance with another aspect of the present invention, at least fiveweighted cross terms, at least ten weighted cross terms, at leasttwenty-five weighted cross terms, at least fifty weighted cross terms,or at least one hundred weighted cross terms, can be used. There is nospecific upper limit on the number of weighted cross terms that can beused with the present invention. More weighted cross terms provide asmoother curve as illustrated in FIG. 2.

For each of the at least two polymer training samples and the at leastone polymer test sample, the area under the respective curve for eachsample is determined. For example, the area under the curve in FIG. 2for the exemplary copolymer is determined. The respective areas underthe curve for each of the at least two polymer training samples are thencorrelated with the respective value of the physical or chemicalproperty or properties of interest. In one aspect of the presentinvention, the area under each curve is divided by 100,000 to calculateprimary structure parameter 1 (PSP1). FIG. 3 illustrates a correlationof a specific physical or chemical property (in this case, Log PENT at2.4 MPa) versus PSP1 for two exemplary series of polyolefin polymers.The results indicate that there is a linear relationship between thelogarithm of the property of interest and the PSP1 value, which isimpacted by the branch content of the high molecular weight fraction ofthe copolymer.

By applying a correlation, such as illustrated in FIG. 3, to therespective area under the curve of the at least one polymer test sample,the value of the physical or chemical property of interest for the atleast one polymer test sample can be determined. This method is furtherillustrated in Examples 1-2 that follow.

Another aspect of the present invention provides a method of determininga value of a physical or chemical property of at least one polymer testsample, the at least one polymer test sample having a molecular weight,a weight fraction at the respective molecular weight, and a number ofshort chain branches per 1000 carton atoms at the respective molecularweight. These parameters can be determined from a MWD profile and a SCBDvia any analytical technique known to one of ordinary skill in the art,such as SEC-FTIR. This method of determining a value of a physical orchemical property of the at least one polymer test sample comprises:

a) providing at least two polymer training samples, each training samplehaving a molecular weight, a weight fraction at the respective molecularweight, a number of short chain branches per 1000 carton atoms at therespective molecular weight, and a known value of the respectivephysical or chemical property;

b) determining a weighted cross term at the respective molecular weightfor each of the at least two polymer training samples and the at leastone polymer test sample, each weighted cross term being determined asthe multiplication product of:

-   -   (1) the respective molecular weight;    -   (2) the weight fraction at the respective molecular weight; and    -   (3) the number of short chain branches per 1000 carbon atoms at        the respective molecular weight;

c) correlating the respective weighted cross terms for each of the atleast two polymer training samples in step b) with the known value ofthe respective physical or chemical property; and

d) applying the correlation of step c) to the weighted cross term of theat least one polymer test sample to determine the value of the physicalor chemical property of the at least one polymer test sample.

The above method is a single-point determination of the methodillustrated in FIGS. 1-3. In this method, a single weighted cross termat a single molecular weight is determined for each of the at least twopolymer training samples and the at least one polymer test sample. Thismethod is further illustrated by Example 3 that follows.

It is apparent from FIG. 3 that not all polymer types have the samecorrelation between the physical or chemical property and the weightedcross term or PSP1. For instance, each family of polyolefin resinsproduced with different catalyst systems (such as chromium,Ziegler-Natta, metallocene, and the like, or combinations thereof) orwith different production processes (such as slurry, solution, gasphase, and the like, or combinations thereof) may have a differentcalibration curve, or a different relationship with the physical orchemical property of interest.

In accordance with a further aspect of the present invention, achemometric method can be used to determine a value of a physical orchemical property of at least one polymer test sample, the at least onepolymer test sample having a MWD profile and a SCBD. This methodcomprises:

a) providing at least two polymer training samples, each training samplehaving a MWD profile, a SCBD, and a known value of the respectivephysical or chemical property;

b) determining at least two weighted cross terms at respective molecularweights along the MWD profile and the SCBD, for each of the at least twopolymer training samples and the at least one polymer test sample, eachweighted cross term being determined as the multiplication product of:

-   -   (1) the respective molecular weight;    -   (2) the weight fraction at the respective molecular weight; and    -   (3) the number of short chain branches per 1000 carbon atoms at        the respective molecular weight;

c) by chemometric analysis, defining a mathematical relationship betweenthe values of the respective physical or chemical property and theweighted cross terms for each of the at least two polymer trainingsamples; and

d) applying the mathematical relationship of step c) to the respectiveweighted cross terms of step b) for the at least one polymer test sampleto determine the value of the physical or chemical property of the atthe least one polymer test sample.

This chemometric method can be used when the at least two polymertraining samples and the at least one polymer test sample are preparedusing catalyst systems that are the same or are different. For example,this method can be used for different families of polyolefin resinsproduced with different catalyst systems (such as chromium,Ziegler-Natta, metallocene, and the like, or combinations thereof) orwith different production processes (such as slurry, solution, gasphase, and the like, or combinations thereof).

Chemometric analysis was described also in U.S. Pat. No. 6,632,680 andU.S. patent application Ser. No. 10/463,849. In the present invention,chemometric analysis can be used to define a mathematical relationshipbetween the values of the respective physical or chemical properties andthe weighted cross terms for each of the at least two polymer trainingsamples. At least two polymer training samples are used in this aspectof the present invention. Alternatively, however, at least threetraining samples, at least four training samples, at least five trainingsamples, at least ten training samples, at least fifteen trainingsamples, or at least twenty training samples, can be used. There is nospecific upper limit on the number of different training samples thatcan be used in this aspect of the present invention. This chemometricmethod is further illustrated by Examples 4-5 that follow.

Primary Structure Parameter 2 (PSP2)

In another aspect, the present invention provides a method ofdetermining a value of a physical or chemical property of at least onepolymer test sample, the at least one polymer test sample having acomposite density, a MWD profile, and a SCBD. The composite density canbe determined via any analytical technique known to one of ordinaryskill in the art, as mentioned above. The method of determining a valueof a physical or chemical property of the at least one polymer testsample comprises:

a) providing at least two polymer training samples, each training samplehaving a composite density, a MWD profile, a SCBD, and a known value ofthe respective physical or chemical property;

b) determining at least two density terms at respective molecularweights along the MWD profile and the SCBD, for each of the at least twopolymer training samples and the at least one polymer test sample, eachdensity term being determined using the composite density, the MWDprofile, and the SCBD;

c) determining a respective melting temperature from each density termin step b);

d) determining a respective probability for tie molecule formation fromeach melting temperature in step c);

e) determining a respective weighted tie molecule probability, eachweighted tie molecule probability being determined as the multiplicationproduct of:

-   -   (1) the weight fraction at the respective molecular weight; and    -   (2) the probability for tie molecule formation in step d) at the        respective molecular weight;

f) plotting each weighted tie molecule probability versus the logarithmof the respective molecular weight for each of the at least two polymertraining samples and the at least one polymer test sample;

g) determining the respective area under each curve in step f);

h) correlating the respective areas for each of the at least two polymertraining samples in step g) with the known value of the respectivephysical or chemical property; and

i) applying the correlation of step h) to the respective area of the atleast one polymer test sample to determine the value of the physical orchemical property of the at least one polymer test sample.

While not intending to be bound by this theory, Applicants believe thathaving more branching on the longer chains—the higher molecular weightfraction—forces these chains into the amorphous region of the polymerand thus increases the probability that they will act as tie molecules,holding the semi-crystalline polymer matrix together. These tiemolecules can contribute significantly to improved toughness properties(e.g., tear, impact, puncture, or stress crack resistance) of polymers.In this aspect of the present invention, the above method is directedtoward determining the value of a physical or chemical property of apolymer test sample using tie molecule probabilities.

This method of the present invention employs at least two differentpolymer training samples. Each polymer training sample has a knowncomposite density, a known MWD profile, and a known SCBD. As notedpreviously, the composite density can be determined via ASTM D 1238,while the MWD profile and the SCBD can be determined using SEC-FTIR, forexample. For an exemplary polyolefin copolymer, FIG. 1 illustrates theMWD profile and the SCBD of the copolymer. Graphical data as illustratedin FIG. 1 can be provided for both the at least two polymer trainingsamples and the at least one polymer test sample. Further, each polymertraining sample has a known value of the respective physical or chemicalproperty that is of interest. For instance, at least two polymertraining samples can be provided that each have a known value of PENT at2.4 MPa, the PENT value having been determined previously using therespective analytical test for PENT.

At least two polymer training samples are used in the methods of thepresent invention. Alternatively, however, at least three trainingsamples, at least four training samples, at least five training samples,at least ten training samples, at least fifteen training samples, or atleast twenty training samples, can be used. There is no specific upperlimit on the number of different training samples that can be used withthe present invention. It is beneficial that the training samplesresemble the polymer test sample, and that the range covered by thetraining samples encompass the polymer test sample, but this is notrequired. Further, the training samples can include duplicate orredundant samples to include the impact of experimental error in thetests for composite density, MWD profile, and SCBD, and of therespective physical or chemical property of interest.

For each of the at least two polymer training samples and the at leastone polymer test sample, at least two density terms at respectivemolecular weights along the MWD profile and the SCBD are determinedusing the composite density, the MWD profile, and the SCBD. In order todetermine the at least two density terms, a relationship between densityand molecular weight is utilized.

The composite density of a polymer can depend on, among other things,the MWD profile and the SCBD of the polymer. In one aspect of thepresent invention, each density term is determined using an empiricalcorrelation between composite density and molecular weight. For certainhomopolymers such as, for example, high density polyethylene, thecomposite density decreases as the molecular weight increases. Using aset of narrow MWD homopolymers (polydispersity index of about 2.3) asdisclosed in Jordens et al. in POLYMER, 41 (2000), 7175, an empiricalcorrelation between composite density and molecular weight can bedetermined. FIG. 4 illustrates a plot of composite density versus thelogarithm of the weight-average molecular weight for an exemplaryhomopolymer, high density polyethylene. By applying this linearcorrelation between composite density and the logarithm of molecularweight to the exemplary MWD profile illustrated in FIG. 1, density termsat respective molecular weights can be determined.

For copolymers with short chain branches or side chains which canfurther suppress density, a correction can be applied to the correlationof FIG. 4. The density terms derived from FIG. 4 are added based ontheir respective weight fraction to estimate a calculated densityassuming no short chain branches. For example, assuming a copolymer witha composite density of about 0.951 g/mL, the calculated density assumingno short chain branches can be about 0.957 g/mL, a difference of about0.006 g/mL. A correction factor based on this change in the compositedensity divided by the average number of SCB's in the SCBD can then beapplied to determine each density term for a copolymer The averagenumber of SCB's can be determined using analytical techniques such asNMR or SEC-FTIR. For the exemplary copolymer in FIG. 1, the averagenumber of SCB's across the whole polymer is about 1.5. Thus, on average,the density term at each molecular weight for this exemplary copolymerdecreases about 0.004 g/mL (0.006 divided by 1.5) for each short chainbranch per 1000 carbon atoms. For simplicity, this assumes, although notcorrect, that each short chain branch suppresses density equally at alllevels of SCB's and at all molecular weights across the MWD profile.This process is illustrated in FIG. 5 for the MWD profile of FIG. 1. Thetop line overlaying the MWD profile in FIG. 5 uses the densitycorrelation of FIG. 4, and assumes homopolymer, i.e., no short chainbranches. The lower line uses the correction factor based on the changein composite density divided by the average number of SCB's in the SCBDto determine density terms across the MWD profile for the copolymer.From FIG. 5, a copolymer density term at each respective MW can bedetermined.

In accordance with one aspect of the present invention, at least twodensity terms at respective molecular weights along the MWD profile andthe SCBD are required. Although at least density two terms are required,it is beneficial to have many more density terms across the range ofmolecular weights to form the curves and correlations illustrated inFIG. 5. Hence, in accordance with another aspect of the presentinvention, at least five density terms, at least ten density terms, atleast twenty-five density terms, at least fifty density terms, or atleast one hundred density terms, can be used. There is no specific upperlimit on the number of density terms that can be used with the presentinvention. More density terms can provide a more accurate correctionfactor and a more accurate copolymer density curve as illustrated inFIG. 5.

A respective melting temperature can then be determined from eachdensity term. One such method is the use of an empirical correlationbetween melting temperature and density, such as that illustrated inFIG. 6 for a polyethylene polymer. The data in this plot is from Patelet al. in J. APPL. POLY. SCI, 60 (1996), 749; Mirabella et al. in J.POLY. SCI., PART B: POLYMER PHYSICS, 40 (2002), 1637; and Huang et al.in J. POLY. SCI., PART B: POLYMER PHYSICS, 28 (1990), 2007. For curvefitting purposes, a point at a density of 1.01 g/mL was assigned by theApplicants. Using FIG. 6, a respective melting temperature can bedetermined from each density term at a respective molecular weightacross the MWD profile of FIG. 5.

In one aspect of the present invention, from each melting temperature isdetermined a respective probability for tie molecule formation. Onetechnique involves determining, at each respective MW, the crystallinelamella thickness (L_(c)) and the amorphous layer thickness (L_(a))using the melting temperature in FIG. 6 and the Gibbs-Thompson equation,which is readily known to one or ordinary skill in the art. Theobjective is not solely to determine L_(c) and L_(a), rather todetermine 2L_(c)+L_(a) at each respective MW. 2L_(c)+L_(a) is generallyunderstood to be the minimum molecule length for a tie molecule, whereinthe tie molecule spans the amorphous layer and spans two crystallinelamella. FIG. 7 illustrates that the method of the present invention todetermine 2L_(c)+L_(a) (measured in nm) as a function of density fitsreasonably well with values of 2L_(c)+L_(a) reported by in theliterature from Patel et al., Mirabella et al., and Huang et al. Somedeviation of the model's prediction can be due to fact that the reportedvalues are at a molecular weight equal to M_(w).

Since 2L_(c)+L_(a) have been determined at each respective MW, theprobability for tie molecule formation can be determined (abbreviated asP or P_(TM)). At each respective MW, P_(TM) is the probability that amolecule will span a distance greater than 2L_(c)+L_(a) at thatrespective MW. Huang et al., in J. MATERIAL SCI., 23 (1988), 3648,described a method to calculate the probability of a molecule with aparticular molecular weight (M_(w)) to form a tie molecule by traversinga distance 2L_(c)+L_(a), using the following equation:

$P = {\frac{1}{3}\frac{\int_{L}^{\infty}{r^{2}{\exp\left( {{- b^{2}}r^{2}} \right)}{\mathbb{d}r}}}{\int_{0}^{\infty}{r^{2}{\exp\left( {{- b^{2}}r^{2}} \right)}{\mathbb{d}r}}}}$${{where}\mspace{14mu} b^{2}} = {{\frac{3}{2{\overset{\_}{r}}^{2}}\mspace{14mu}{and}\mspace{14mu}{\overset{\_}{r}}^{2}} = {{\left( {Dnl}^{2} \right).{The}}\mspace{14mu}{symbols}\mspace{14mu}{above}\mspace{14mu}{have}\mspace{14mu}{the}\mspace{14mu}{following}\mspace{14mu}{meanings}\text{:}}}$P = Probability  of  tie-chain  formationL = Critical  distance = 2L_(c) + L_(a) L_(c) = Lamella  thicknessL_(a) = Amorphous  layer  thicknessD = Chain  extension  factor  in  melt = 6.8  for  polyethylenen = Number  of  links = M_(w)/14  for  polyethylenel = The  link  length = 0.153  nm  for  polyethylene.

Spreadsheet and/or computer based methods can be used to determine thevalue of P or P_(TM) at each respective MW from the respective meltingtemperature. Calculated values of P_(TM) using the present methodcompare well with values reported in the literature, such as by Patel etal. Some limitations of this method for determining P_(TM) can includethat P_(TM) alone does not reflect the actual tie-molecule concentrationin semi-crystalline polymers (i.e. loops from entanglements may serve asjunction points as well). Moreover, only static tie molecule levels areaccounted for in these calculations and do not include new tie chains(dynamic) that can form due to lamellar sliding as a result ofdeformation.

For each of the at least two polymer training samples and the at leastone polymer test sample, weighted tie molecule probabilities can bedetermined using the respective data for each sample. A weighted tiemolecule probability is the multiplication product of a weight fractionat a respective molecular weight and P_(TM), the probability for tiemolecule formation, at the respective molecular weight (i.e., dW/d(LogM)*P_(TM)).

In accordance with one aspect of the present invention, at least twoweighted tie molecule probabilities at respective molecular weightsalong the MWD profile and the SCBD are required. Each weighted tiemolecule probability is plotted versus the logarithm of the respectivemolecular weight, for each of the at least two polymer training samplesand the at least one polymer test sample. This is illustrated in FIG. 8for two exemplary polyolefin copolymers. Although at least two weightedtie molecule probabilities are required, it is beneficial to have manymore probability terms across the range of molecular weights to form thecurve illustrated in FIG. 8. Hence, in accordance with another aspect ofthe present invention, at least five weighted tie moleculeprobabilities, at least ten weighted tie molecule probabilities, atleast twenty-five weighted tie molecule probabilities, at least fiftyweighted tie molecule probabilities, or at least one hundred weightedtie molecule probabilities, can be used. There is no specific upperlimit on the number of weighted tie molecule probabilities that can beused with the present invention. More weighted tie moleculeprobabilities provide a smoother curve as illustrated in FIG. 8.

For each of the at least two polymer training samples and the at leastone polymer test sample, the area under the respective curve for eachsample is determined. For example, the areas under the curves in FIG. 8for the exemplary copolymers are determined. The respective areas underthe curve for each of the at least two polymer training samples are thencorrelated with the respective value of the physical or chemicalproperty or properties of interest. In one aspect of the presentinvention, the area under each curve is multiplied by 100 to calculateprimary structure parameter 2 (PSP2). In FIG. 8, the polymer sample withthe higher peak has a PSP2 value of 10.9, while the polymer sample withthe smaller peak has a PSP2 value of 9.8. Correlations with a specificphysical or chemical property using the at least two polymer trainingsamples also can made with the PSP2 value.

By applying the correlation with the respective areas under the curvesof the at least two polymer training samples to the respective areaunder the curve of the at least one polymer test sample, the value ofthe physical or chemical property of interest for the at least onepolymer test sample can be determined. This method is furtherillustrated in Examples 6-8 that follow.

This method involving tie molecules can be used when the at least twopolymer training samples and the at least one polymer test sample areprepared using catalyst systems that are the same or are different. Forexample, this method can be used for different families of polyolefinresins produced with different catalyst systems (such as chromium,Ziegler-Natta, metallocene, and the like, or combinations thereof) orwith different production processes (such as slurry, solution, gasphase, and the like, or combinations thereof).

Another aspect of the present invention provides a method of determininga value of a physical or chemical property of at least one polymer testsample, the at least one polymer test sample having a composite density,a molecular weight, and a weight fraction at the respective molecularweight. These parameters can be determined via various analyticaltechniques known to one of ordinary skill in the art, as discussedpreviously. This method of determining a value of a physical or chemicalproperty of the at least one polymer test sample comprises:

a) providing at least two polymer training samples, each training samplehaving a composite density, a molecular weight, a weight fraction at therespective molecular weight, and a known value of the respectivephysical or chemical property;

b) determining a minimum molecule length for a tie molecule(2L_(c)+L_(a)) using the composite density for each of the at least twopolymer training samples and the at least one polymer test sample;

c) determining a respective probability for tie molecule formation atthe respective molecular weight from each 2L_(c)+L_(a) of step b);

d) determining a respective weighted tie molecule probability, eachweighted tie molecule probability being determined as the multiplicationproduct of:

-   -   (1) the weight fraction at the respective molecular weight; and    -   (2) the probability for tie molecule formation in step c) at the        respective molecular weight;

e) correlating the respective weighted tie molecule probability for eachof the at least two polymer training samples in step d) with the knownvalue of the respective physical or chemical property; and

f) applying the correlation of step e) to the weighted tie moleculeprobability of the at least one polymer test sample to determine thevalue of the physical or chemical property of the at least one polymertest sample.

The above method is a single-point determination of the methodillustrated in FIGS. 4-8. In this method, a single weighted tie moleculeprobability is determined at a single molecular weight for each of theat least two polymer training samples and the at least one polymer testsample. In this aspect, the composite density reflects the impact ofshort chain branching and molecular weight across the whole polymer.Using the correlation illustrated in FIG. 7, respective 2L_(c)+L_(a)values can be determined from the respective composite density of thepolymer sample. This method is further illustrated in Example 9 thatfollows.

EXAMPLES Example 1

Table I lists the Mw, composite density, and measured PENT (hours) at2.4 MPa for seven (7) bimodal polyethylene polymers produced usingZiegler-Natta catalysts. FIGS. 9-15 are the MWD profile and SCBD ofthese seven polymer training samples. These figures illustrate themeasured data and the fitted SCBD using an SEC-FTIR technique, asdescribed in U.S. Pat. No. 6,632,680 and U.S. patent application Ser.No. 10/463,849. Due to the low weight fraction at very high molecularweights, it is often difficult to accurately measure the SCB's at theserespective molecular weights. Application of the methodology discussedearlier relative to the weighted cross terms of these seven trainingsamples resulted in the curves illustrated in FIG. 16. The areas underthe curves in FIG. 16 decrease from polyethylene polymer resin BM-1 toBM-7. FIG. 17 illustrates that a strong linear correlation existsbetween the Log PENT value and PSP1 for these polymers. Since an exactPENT value was not established for the BM-1 polymer, it was not plottedon FIG. 17. Thus, the PENT value of a test sample (an unknown or anexperimental sample) from the same bimodal polymer family could bedetermined by applying the correlation in FIG. 17 to the PSP1 value ofthe test sample.

Example 2

Five (5) unimodal polyethylene polymers were produced using achromium-based catalyst. These polymers had composite densities in therange of about 0.948 to about 0.950 and M_(w) in the range of about 260to about 320 kg/mol. The PENT value at 2.4 MPa, the MWD profile, and theSCBD of these five polymer training samples were also provided. Althoughthe MWD profile and SCBD data are not shown, this information isexemplified in FIGS. 9-15 related to Example 1. Application of themethodology discussed earlier relative to the weighted cross terms ofthese five training samples resulted in the curves illustrated in FIG.18. FIG. 19 illustrates that a strong linear correlation exists betweenthe Log PENT value and PSP1 for these polymers. Thus, the PENT value ofa test sample from the same polymer family could be determined byapplying the correlation in FIG. 19 to the PSP1 value of the testsample.

Example 3

Example 3 uses the same bimodal polyethylene polymers discussed inExample 1. Table II lists a specific molecular weight, the weightfraction at that molecular weight, the SCB's at that molecular weight,and the measured PENT (hours) at 2.4 MPa for six (6) bimodalpolyethylene polymers. Weighted cross terms were calculated for each ofthese six training samples, divided by 100,000, and plotted against themeasured PENT value. Even with this single point method (i.e., takingdata only at one molecular weight, Log M=6.05), FIG. 20 illustrates thata strong linear correlation exists between the Log PENT value and theweighted cross term divided by 100,000. Thus, the PENT value of a testsample from the same bimodal polymer family could be determined byapplying the correlation in FIG. 20 to a single weighted cross term ofthe test sample.

Example 4

Table III lists the predicted PENT value at 2.4 MPa using chemometricanalysis versus the measured PENT value for a series of polyethylenepolymers made using chromium-based, Ziegler-Natta, and dual catalystsystems. This data is illustrated graphically in FIG. 21. The MWDprofile and SCBD of these polymer training samples were provided viaSEC-FTIR. Although the MWD profile and SCBD data are not shown, thisinformation is exemplified in FIGS. 9-15 related to Example 1.Respective weighted cross terms along the MWD profile and the SCBD weredetermined for each of the polymer training samples. Chemometric methodswere used to generate a mathematical relationship, or correlation,between PENT values and the weighted cross terms, and subsequently toPSP1. The mathematical relationship illustrated in FIG. 21 wasindependent of the catalyst system and process used to produce thepolymer training samples. Thus, the PENT value of a test sample could bedetermined by applying the chemometric analysis to the PSP1 value of thetest sample.

Example 5

Table IV lists the predicted PENT value at 2.4 MPa using chemometricanalysis versus the measured PENT value for a series of polyethylenepolymers made using different catalyst systems. This data is illustratedgraphically in FIG. 22. The MWD profile and SCBD of these polymertraining samples were provided via SEC-FTIR. Although the MWD profileand SCBD data are not shown, this information is exemplified in FIGS.9-15 related to Example 1. Respective weighted cross terms along the MWDprofile and the SCBD were determined for each of the polymer trainingsamples. Chemometric methods were used to generate a mathematicalrelationship, or correlation, between PENT values and the weighted crossterms, and subsequently to PSP1. The mathematical relationshipillustrated in FIG. 22 was independent of the catalyst system andprocess used to produce the polymer training samples. Thus, the PENTvalue of a test sample could be determined by applying the chemometricanalysis to the PSP1 value of the test sample.

Example 6

In this example, two different polymer families were evaluated. Five (5)unimodal polyethylene polymers produced using a chromium-based catalystwere selected. These polymers had a broad MWD; the averagepolydispersity index for the five polymers was about 30. Six (6) bimodalpolyethylene polymers produced using Ziegler-Natta catalysts wereselected. These bimodal polymers had an average polydispersity index ofabout 17. The PENT value at 2.4 MPa, the MWD profile, and the SCBD ofthese eleven polymer training samples were provided. Although the MWDprofile and SCBD data are not shown, this information is exemplified inFIGS. 9-15 related to Example 1. Composite densities were provided andranged from about 0.947 to about 0.957 g/mL for the eleven polymertraining samples. Application of the methodology discussed earlierrelative to the weighted tie molecule probabilities of these eleventraining samples resulted in curves similar to those exemplified in FIG.8. The respective areas under the curves were determined and multipliedby 100 to calculate the respective PSP2 value for each training sample.FIG. 23 illustrates that a strong linear correlation exists between theLog PENT value and PSP2 for these polymers, irrespective of the catalystsystem used to produce the polymer training samples. Thus, the PENTvalue of a polymer test sample produced using similar or differentcatalyst systems could be determined by applying the correlation in FIG.23 to the PSP2 value of the test sample.

Example 7

In this example, three different polymer families were evaluated:unimodal polyethylene polymers produced using a chromium-based catalystand polymers produced using Ziegler-Natta catalysts, and bimodalpolyethylene polymers produced using Ziegler-Natta catalysts. TheSP-NCTL value, the MWD profile, and the SCBD of these fifteen (15)polymer training samples were provided. Although the MWD profile andSCBD data are not shown, this information is exemplified in FIGS. 9-15related to Example 1. Composite densities were provided and ranged fromabout 0.93 to about 0.96 g/mL for the fifteen polymer training samples.Application of the methodology discussed earlier relative to theweighted tie molecule probabilities of these eleven training samplesresulted in curves similar to those exemplified in FIG. 8. Therespective areas under the curves were determined and multiplied by 100to calculate the respective PSP2 value for each training sample. FIG. 24illustrates that a strong linear correlation exists between the LogSP-NCTL value and PSP2 for these copolymers, irrespective of thecatalyst system used to produce the polymer training sample. Further,these polymers spanned a large composite density range. Thus, theSP-NCTL value of a polymer test sample produced using similar ordifferent catalyst systems could be determined by applying thecorrelation in FIG. 24 to the PSP2 value of the test sample.

Example 8

In this example, five different polymer families were evaluated:unimodal polyethylene polymers produced using a chromium-based catalyst,Ziegler-Natta catalysts, and metallocene catalysts; and bimodalpolyethylene polymers produced using Ziegler-Natta catalysts andmetallocene catalysts. Typical applications for these polymer resinsinclude, but are not limited to, blow molding, pipe, geomembrane, andfilm applications. The NDR value, the MWD profile, and the SCBD of thesethirty-eight (38) polymer training samples were provided. Although theMWD profile and SCBD data are not shown, this information is exemplifiedin FIGS. 9-15 related to Example 1. Composite densities were providedand ranged from about 0.91 to about 0.96 g/mL for the thirty-eight (38)polymer training samples. Application of the methodology discussedearlier relative to the weighted tie molecule probabilities of thesethirty-eight training samples resulted in curves similar to thoseexemplified in FIG. 8. The respective areas under the curves weredetermined and multiplied by 100 to calculate the respective PSP2 valuefor each training sample. FIG. 25 illustrates that a strong linearcorrelation exists between the NDR value and PSP2 for these polymers,irrespective of the catalyst system used to produce the polymer trainingsamples. Further, these polymers spanned a large composite density rangeand are used in a wide variety of end-use applications. Thus, the NDRvalue of a polymer test sample produced using similar or differentcatalyst systems could be determined by applying the correlation in FIG.25 to the PSP2 value of the test sample.

Example 9

Example 1 uses the same bimodal polyethylene polymers discussed inExample 1. Table V lists a specific molecular weight, the weightfraction at that molecular weight, the composite density and themeasured PENT (hours) at 2.4 MPa for six (6) bimodal polyethylenepolymers. 2L_(c)+L_(a) values were determined and weighted tie moleculeprobabilities were calculated for each of these six training samples.The Log PENT value was plotted against the weighted tie moleculeprobabilities, as shown in FIG. 26. Even with this single point method(i.e., taking data only at one molecular weight, Log M=5.7), FIG. 26illustrates that a strong linear correlation exists between the Log PENTvalue and the weighted tie molecule probability. Thus, the PENT value ofa test sample from the same bimodal polymer family could be determinedby applying the correlation in FIG. 26 to a single weighted tie moleculeprobability of the test sample.

TABLE I Bimodal polyethylene polymer data. M_(w) Density PENT (h) Resin(kg/mol) (g/cm³) (2.4 MPa) BM-1 320 0.947 >6000 BM-2 290 0.949 3028 BM-3291 0.951 1046.5 BM-4 270 0.951 625.5 BM-5 260 0.953 406 BM-6 228 0.95436.5 BM-7 193 0.959 3

TABLE II Bimodal polyethylene polymer data at a single molecular weight.Resin Log M dW/dLogM SCB's PENT(h) @ 2.4 MPa BM-2 6.05 0.201 2.96 3028BM-3 6.05 0.197 2.76 1046 BM-4 6.05 0.174 2.65 625 BM-5 6.05 0.166 2.65406 BM-6 6.05 0.146 2.63 36 BM-7 6.05 0.122 2.61 3

TABLE III Predicted PENT (h) using Chemometric Analysis versus MeasuredPENT (h). Measured Predicted PENT PENT CrO-1 1 8 CrO-1 2 8 CrO-1 5 7CrO-1 10 6 DCE-1 36 167 DCE-1 36 167 DCE-1 36 167 CrO-2 379 443 CrO-2379 241 CrO-2 379 415 CrO-3 508 142 CrO-3 508 591 CrO-4 838 812 CrO-51400 1415 CrO-6 1872 2330 BM-1 2000 1791 BM-2 2000 2316 CrO-7 3000 2468DCE-2 3000 3136 DCE-3 3500 3376 DCE-3 3500 3376

TABLE IV Predicted PENT (h) using Chemometric Analysis versus MeasuredPENT (h). Measured Predicted PENT(h) PENT(h) BM-A 3028 3034 BM-B 28773109 BM-C 3172 2963 BM-D 1047 1001 BM-E 1047 1001 BM-F 1047 1001 BM-G 37192 BM-H 35 192 BM-I 38 192 BM-J 406 469 BM-K 386 474 BM-L 425 465 BM-M3 −184 BM-N 3 −184 BM-O 3 −184 CR-A 77 97 CR-B 69 100 CR-C 84 93 CR-D374 483 CR-E 337 499 CR-F 411 466 CR-G 506 590 CR-H 455 615 CR-I 556 564CR-J 839 691 CR-K 755 729 CR-L 923 652 CR-M 1395 1344 CR-N 1255 1413CR-O 1534 1274

TABLE V Bimodal polyethylene polymer data at a single molecular weight.Resin Log M dW/dLogM Density PENT(h) @ 2.4 MPa BM-2 5.7 0.350 0.94933028 BM-3 5.7 0.336 0.9509 1046 BM-4 5.7 0.317 0.9514 625 BM-5 5.7 0.3030.9527 406 BM-6 5.7 0.268 0.9544 36 BM-7 5.7 0.227 0.9594 3

1. A method of determining a value of a physical or chemical property ofat least one polymer test sample, the at least one polymer test samplehaving a molecular weight distribution (MWD) profile and a short chainbranching distribution (SCBD), comprising: a) providing at least twopolymer training samples, each training sample having a MWD profile, aSCBD, and a known value of the respective physical or chemical property;b) determining at least two weighted cross terms at respective molecularweights along the MWD profile and the SCBD, for each of the at least twopolymer training samples and the at least one polymer test sample, eachweighted cross term being determined as the multiplication product of:(1) the respective molecular weight; (2) the weight fraction at therespective molecular weight; and (3) the number of short chain branchesper 1000 carbon atoms at the respective molecular weight; c) plottingeach weighted cross term versus the logarithm of the respectivemolecular weight for each of the at least two polymer training samplesand the at least one polymer test sample; d) determining the respectivearea under each curve in step c); e) correlating the respective areasfor each of the at least two polymer training samples in step d) withthe known value of the respective physical or chemical property; and f)applying the correlation of step e) to the respective area of the atleast one polymer test sample to determine the value of the physical orchemical property of the at least one polymer test sample.
 2. The methodof claim 1, wherein the physical or chemical property is PENT value at2.4 MPa.
 3. The method of claim 1, wherein the physical or chemicalproperty is PENT value at 3.8 MPa.
 4. The method of claim 1, wherein thephysical or chemical property is Environmental Stress Crack Resistance(ESCR).
 5. The method of claim 1, wherein the physical or chemicalproperty is Single Point Notched Constant Tensile Load (SP-NCTL).
 6. Themethod of claim 1, wherein the physical or chemical property is NotchedPipe Test (NPT).
 7. The method of claim 1, wherein the physical orchemical property is Full Notched Creep Test (FNCT).
 8. The method ofclaim 1, wherein the physical or chemical property is Natural Draw Ratio(NDR).
 9. The method of claim 1, wherein the MWD profile and the SCBDare obtained from SEC-FTIR analysis.
 10. The method of claim 9, whereinless than about 1 gram each of the at least two polymer training samplesand the at least one polymer test sample is provided for analysis. 11.The method of claim 1, wherein the area under the curve in step d) isdivided by 100,000 to calculate primary structure parameter 1 (PSP1).12. A method of determining a value of a physical or chemical propertyof at least one polymer test sample, the at least one polymer testsample having a MWD profile and a SCBD, comprising: a) providing atleast two polymer training samples, each training sample having a MWDprofile, a SCBD, and a known value of the respective physical orchemical property; b) determining at least two weighted cross terms atrespective molecular weights along the MWD profile and the SCBD, foreach of the at least two polymer training samples and the at least onepolymer test sample, each weighted cross term being determined as themultiplication product of: (1) the respective molecular weight; (2) theweight fraction at the respective molecular weight; and (3) the numberof short chain branches per 1000 carbon atoms at the respectivemolecular weight; c) by chemometric analysis, defining a mathematicalrelationship between the values of the respective physical or chemicalproperty and the weighted cross terms for each of the at least twopolymer training samples; and d) applying the mathematical relationshipof step c) to the respective weighted cross terms of step b) for the atleast one polymer test sample to determine the value of the physical orchemical property of the at least one polymer test sample.
 13. Themethod of claim 12, wherein the physical or chemical property is PENTvalue at 2.4 MPa.
 14. The method of claim 12, wherein the physical orchemical property is PENT value at 3.8 MPa.
 15. The method of claim 12,wherein the physical or chemical property is Environmental Stress CrackResistance (ESCR).
 16. The method of claim 12, wherein the physical orchemical property is Single Point Notched Constant Tensile Load(SP-NCTL).
 17. The method of claim 12, wherein the physical or chemicalproperty is Notched Pipe Test (NPT).
 18. The method of claim 12, whereinthe physical or chemical property is Full Notched Creep Test (FNCT). 19.The method of claim 12, wherein the physical or chemical property isNatural Draw Ratio (NDR).
 20. The method of claim 12, wherein the MWDprofile and the SCBD are obtained from SEC-FTIR analysis.
 21. The methodof claim 20, wherein less than about 1 gram each of the at least twopolymer training samples and the at least one polymer test sample isprovided for analysis.
 22. The method of claim 12, wherein the at leasttwo polymer training samples and the at least one polymer test sampleare prepared using catalyst systems that are the same or are different.23. A method of determining a value of a physical or chemical propertyof at least one polymer test sample, the at least one polymer testsample having a composite density, a MWD profile, and a SCBD,comprising: a) providing at least two polymer training samples, eachtraining sample having a composite density, a MWD profile, a SCBD, and aknown value of the respective physical or chemical property; b)determining at least two density terms at respective molecular weightsalong the MWD profile and the SCBD, for each of the at least two polymertraining samples and the at least one polymer test sample, each densityterm being determined using the composite density, the MWD profile, andthe SCBD; c) determining a respective melting temperature from eachdensity term in step b); d) determining a respective probability for tiemolecule formation from each melting temperature in step c); e)determining a respective weighted tie molecule probability, eachweighted tie molecule probability being determined as the multiplicationproduct of: (1) the weight fraction at the respective molecular weight;and (2) the probability for tie molecule formation in step d) at therespective molecular weight; f) plotting each weighted tie moleculeprobability versus the logarithm of the respective molecular weight foreach of the at least two polymer training samples and the at least onepolymer test sample; g) determining the respective area under each curvein step f); h) correlating the respective areas for each of the at leasttwo polymer training samples in step g) with the known value of therespective physical or chemical property; and i) applying thecorrelation of step h) to the respective area of the at least onepolymer test sample to determine the value of the physical or chemicalproperty of the at least one polymer test sample.
 24. The method ofclaim 23, wherein the physical or chemical property is PENT value at 2.4MPa.
 25. The method of claim 23, wherein the physical or chemicalproperty is PENT value at 3.8 MPa.
 26. The method of claim 23, whereinthe physical or chemical property is Environmental Stress CrackResistance (ESCR).
 27. The method of claim 23, wherein the physical orchemical property is Single Point Notched Constant Tensile Load(SP-NCTL).
 28. The method of claim 23, wherein the physical or chemicalproperty is Notched Pipe Test (NPT).
 29. The method of claim 23, whereinthe physical or chemical property is Full Notched Creep Test (FNCT). 30.The method of claim 23, wherein the physical or chemical property isNatural Draw Ratio (NDR).
 31. The method of claim 23, wherein the MWDprofile and the SCBD are obtained from SEC-FTIR analysis.
 32. The methodof claim 31, wherein less than about 1 gram each of the at least twopolymer training samples and the at least one polymer test sample isprovided for analysis.
 33. The method of claim 23, wherein the at leasttwo polymer training samples and the at least one polymer test sampleare prepared using catalyst systems that are the same or are different.34. The method of claim 23, wherein the area under the curve in step g)is multiplied by 100 to calculate primary structure parameter 2(PSP2).35. The method of claim 23, wherein each density term of step b) isdetermined using an empirical correlation between composite density andmolecular weight, and a correction factor based on a change in thecomposite density divided by the average number of short chain branchesper 1000 carbon atoms in the SCBD.
 36. The method of claim 23, whereineach melting temperature of step c) is determined using an empiricalcorrelation of melting temperature with density.
 37. A method ofdetermining a value of a physical or chemical property of at least onepolymer test sample, the at least one polymer test sample having amolecular weight, a weight fraction at the respective molecular weight,and a number of short chain branches per 1000 carton atoms at therespective molecular weight, comprising: a) providing at least twopolymer training samples, each training sample having a molecularweight, a weight fraction at the respective molecular weight, a numberof short chain branches per 1000 carton atoms at the respectivemolecular weight, and a known value of the respective physical orchemical property; b) determining a weighted cross term at therespective molecular weight for each of the at least two polymertraining samples and the at least one polymer test sample, each weightedcross term being determined as the multiplication product of: (1) therespective molecular weight; (2) the weight fraction at the respectivemolecular weight; and (3) the number of short chain branches per 1000carbon atoms at the respective molecular weight; c) correlating therespective weighted cross terms for each of the at least two polymertraining samples in step b) with the known value of the respectivephysical or chemical property; and d) applying the correlation of stepc) to the weighted cross term of the at least one polymer test sample todetermine the value of the physical or chemical property of the at leastone polymer test sample.